Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

• Published in: Advances in Mathematical Physics Vol. 2020; no. 2020; pp. 1 - 11
• Main Authors: Tariboon, Jessada, Ntouyas, Sotiris. K., Ahmad, Bashir, Wongcharoen, Athasit
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Publishing Corporation 2020; Hindawi; John Wiley & Sons, Inc; Wiley
• Subjects: Boundary conditions; Boundary value problems; Differential equations; Existence theorems; Fixed points (mathematics); Hypotheses; Mathematical analysis; Uniqueness theorems
• ISSN: 1687-9120; 1687-9139
• DOI: 10.1155/2020/9606428

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.